A good universal weight for multiple recurrence averages with commuting transformations in norm

We will show that the sequences appearing in Bourgain's double recurrence result are good universal weights to the multiple recurrence averages with commuting measure-preserving transformations in norm. This will extend the pointwise converge result of Bourgain, the norm convergence result of Tao, and the authors' previous work on the single measure-preserving transformation. The proof will use the double-recurrence Wiener-Wintner theorem, factor decompositions (Host-Kra-Ziegler factors), nilsequences, and various seminorms including the ones by Gowers-Host-Kra as well as the box seminorms introduced by Host.